The bound of Eq. (REF ) is attainable asymptotically in the limit
\(M\rightarrow \infty \)  as follows from the results in [1]}, [2]}. One
may think that the trace distance has a more natural operational meaning than
the QCB. In spite of this, it does not adapt to the case of many copies;
indeed, one can find states \(\rho ,\sigma ,\rho ^{\prime },\sigma ^{\prime }\)  such
that
\(T(\rho ,\sigma )<T(\rho ^{\prime },\sigma ^{\prime })\quad \hbox{but}\quad T(\rho ^{\prime }{}^{M},\sigma ^{\prime }{}^{M})<T(\rho ^{M},\sigma ^{M})\,.\) 
